Ruffini's rule

Ruffini's rule is a method for computing the Euclidean division of a polynomial by a binomial of the form x minus r. It was described by Paolo Ruffini in 1809 and is a special case of synthetic division in which the divisor is a linear monic factor. The coefficients of the polynomial are written in order of decreasing degree, with zero for any missing term. The leading coefficient is brought down, then each result is multiplied by r, added to the next coefficient and carried on until no numbers remain. The final value is the remainder, which equals the value of the polynomial at r, as the polynomial remainder theorem states, and the other values are the coefficients of the quotient. The rule is commonly used to factor a polynomial once a root is known.